1.Financial Derivatives and Discrete Time Models I

8/3/2019 1.Financial Derivatives and Discrete Time Models I

http://slidepdf.com/reader/full/1financial-derivatives-and-discrete-time-models-i 1/11

1 I n t r o d u c t i o n t o d e r i v a t i v e s e c u r i t i e s

1 . 1 S o m e d e n i t i o n s f r o m n a n c e

T h e r e a r e a n e n o r m o u s n u m b e r o f d e r i v a t i v e s e c u r i t i e s b e i n g t r a d e d i n n a n c i a l

m a r k e t s . R a t h e r t h a n s p e n d i n g t i m e l i s t i n g t h e m h e r e , w e r e f e r t o t h e r e f e r e n c e s

a n d j u s t d e n e t h o s e s e c u r i t i e s t h a t w e s h a l l b e p r i c i n g .

D e n i t i o n 1 . 1 A f o r w a r d c o n t r a c t i s a n a g r e e m e n t t o b u y o r s e l l a n a s s e t o n

a s p e c i e d f u t u r e d a t e f o r a s p e c i e d p r i c e .

F o r w a r d s a r e n o t g e n e r a l l y t r a d e d o n e x c h a n g e s . I t c o s t s n o t h i n g t o e n t e r i n t o

a f o r w a r d c o n t r a c t . A f u t u r e c o n t r a c t i s t h e s a m e a s a f o r w a r d e x c e p t t h a t f u t u r e s

a r e n o r m a l l y t r a d e d o n e x c h a n g e s a n d t h e e x c h a n g e s p e c i e s c e r t a i n s t a n d a r d

f e a t u r e s o f t h e c o n t r a c t .

F o r m o s t o f t h i s c o u r s e , w e s h a l l b e c o n c e r n e d w i t h t h e p r i c i n g o f o p t i o n s . O p -

t i o n s c o m e i n m a n y d i e r e n t t y p e s o f w h i c h t h e m o s t i m p o r t a n t a r e t h e A m e r i c a n

a n d E u r o p e a n o p t i o n s . O p t i o n s w e r e r s t t r a d e d o n a n o r g a n i s e d e x c h a n g e i n

1 9 7 3 , s i n c e w h e n t h e r e h a s b e e n a d r a m a t i c g r o w t h i n t h e o p t i o n s m a r k e t s . A n

o p t i o n i s s o - c a l l e d b e c a u s e i t g i v e s t h e h o l d e r t h e r i g h t , b u t n o t t h e o b l i g a t i o n , t o

d o s o m e t h i n g .

D e n i t i o n 1 . 2 A c a l l o p t i o n g i v e s t h e h o l d e r t h e r i g h t t o b u y . A p u t o p t i o n g i v e s

t h e h o l d e r t h e r i g h t t o s e l l .

A E u r o p e a n c a l l r e s p . p u t o p t i o n g i v e s t h e h o l d e r t h e r i g h t , b u t n o t t h e

o b l i g a t i o n , t o b u y r e s p . s e l l a n a s s e t o n a c e r t a i n d a t e t h e e x p i r a t i o n d a t e ,

e x e r c i s e d a t e o r m a t u r i t y , f o r a c e r t a i n p r i c e t h e s t r i k e p r i c e .

A n A m e r i c a n o p t i o n i s s i m i l a r e x c e p t t h a t t h e h o l d e r o f a n A m e r i c a n c a l l , f o r

e x a m p l e , h a s t h e r i g h t t o b u y t h e a s s e t f o r t h e s p e c i e d p r i c e a t a n y t i m e u p

u n t i l t h e e x p i r a t i o n d a t e . I t c a n b e o p t i m a l t o e x e r c i s e s u c h a n o p t i o n p r i o r t o

e x p i r a t i o n . A m e r i c a n o p t i o n s a r e h a r d e r t o a n a l y s e t h a n E u r o p e a n o n e s a n d w e

s h a l l t o u c h o n t h e m a t m o s t b r i e y .

T h e r e a r e t w o m a i n u s e s f o r o p t i o n s , s p e c u l a t i o n a n d h e d g i n g . I n s p e c u l a t i o n ,

a v a i l a b l e f u n d s a r e i n v e s t e d o p p o r t u n i s t i c a l l y i n t h e h o p e o f m a k i n g a p r o t .

H e d g i n g i s t y p i c a l l y e n g a g e d i n b y c o m p a n i e s w h o h a v e t o d e a l h a b i t u a l l y i n

i n t r i n s i c a l l y r i s k y a s s e t s s u c h a s f o r e i g n e x c h a n g e , w h e a t , c o p p e r , o i l a n d s o o n .

F o r h e d g e r s , t h e b a s i c p u r p o s e o f a n o p t i o n i s t o s p r e a d r i s k .

S u p p o s e t h a t I k n o w t h a t i n t h r e e m o n t h s t i m e I s h a l l n e e d a m i l l i o n g a l l o n s o f

j e t f u e l . I a m w o r r i e d b y t h e u c t u a t i o n s i n t h e p r i c e o f f u e l a n d s o I b u y E u r o p e a n

c a l l o p t i o n s . T h i s w a y I k n o w t h e m a x i m u m a m o u n t o f m o n e y t h a t I s h a l l n e e d

t o b u y t h e f u e l . L e t ' s d e n o t e t h e p r i c e o f t h e u n d e r l y i n g a s s e t j e t f u e l a t t i m e T

w h e n t h e o p t i o n e x p i r e s t h r e e m o n t h s t i m e b y S

T

a n d s u p p o s e t h a t t h e s t r i k e

p r i c e i s K . I f , a t t i m e T , S

T

K t h e n I s h a l l e x e r c i s e t h e o p t i o n . T h e o p t i o n i s

t h e n s a i d t o b e i n t h e m o n e y : I a m a b l e t o b u y a n a s s e t w o r t h S

T

K d o l l a r s f o r

j u s t K d o l l a r s . I f o n t h e o t h e r h a n d S

T

K t h e n I ' l l b u y m y f u e l o n t h e o p e n

m a r k e t a n d t h r o w a w a y t h e o p t i o n w h i c h i s w o r t h l e s s . T h e o p t i o n i s t h e n s a i d

t o b e o u t o f t h e m o n e y . T h e p a y o f r o m t h e o p t i o n i s t h u s m a x S

T

; K , K o r

S

T

, K

+

.



T h e f u n d a m e n t a l p r o b l e m i s t o d e t e r m i n e h o w m u c h I s h o u l d b e w i l l i n g t o p a y

f o r s u c h a n o p t i o n . Y o u c a n t h i n k o f i t a s a n i n s u r a n c e p r e m i u m . I a m i n s u r i n g

m y s e l f a g a i n s t t h e p r i c e o f j e t f u e l g o i n g u p .

I n o r d e r t o a n s w e r t h i s q u e s t i o n w e a r e g o i n g t o h a v e t o m a k e s o m e a s s u m p -

t i o n s a b o u t t h e w a y i n w h i c h m a r k e t s o p e r a t e . T o f o r m u l a t e t h e s e w e b e g i n b y

d i s c u s s i n g f o r w a r d c o n t r a c t s i n m o r e d e t a i l .

1 . 2 P r i c i n g a f o r w a r d

R e c a l l t h a t a f o r w a r d c o n t r a c t i s a n a g r e e m e n t t o b u y o r s e l l a n a s s e t o n a

s p e c i e d f u t u r e d a t e f o r a s p e c i e d p r i c e . T h e p r i c i n g p r o b l e m h e r e i s ` W h a t

p r i c e f o r t h e a s s e t s h o u l d b e s p e c i e d i n t h e c o n t r a c t ? ' .

P r o b l e m 1 . 3 S u p p o s e t h a t I e n t e r i n t o a l o n g p o s i t i o n o n a f o r w a r d c o n t r a c t ,

t h a t i s I a g r e e t o b u y t h e a s s e t f o r p r i c e K a t t i m e T . T h e n t h e p a y o a t t i m e T

i s S

T

, K , w h e r e S

T

i s t h e a s s e t p r i c e a t t i m e T , s i n c e I m u s t b u y a n a s s e t

w o r t h S

T

f o r p r i c e K . T h i s p a y o c o u l d b e p o s i t i v e o r n e g a t i v e a n d s i n c e t h e c o s t

o f e n t e r i n g i n t o a f o r w a r d c o n t r a c t i s z e r o t h i s i s a l s o m y t o t a l g a i n o r l o s s f r o m

t h e c o n t r a c t . O u r q u e s t i o n t h e n i s ` w h a t i s a f a i r v a l u e o f K ? '

A t t h e t i m e w h e n t h e c o n t r a c t i s w r i t t e n , w e d o n ' t k n o w S

T

, w e c a n o n l y g u e s s

a t i t . T y p i c a l l y , w e a s s i g n a p r o b a b i l i t y d i s t r i b u t i o n t o i t . A w i d e l y a c c e p t e d

m o d e l t h a t w e r e t u r n t o i n x 7 i s t h a t s t o c k p r i c e s a r e l o g - n o r m a l l y d i s t r i b u t e d .

T h a t i s , t h e r e a r e c o n s t a n t s a n d s o t h a t t h e l o g a r i t h m o f S

T

= S

0

t h e s t o c k

p r i c e a t t i m e T d i v i d e d b y t h a t a t t i m e z e r o i s n o r m a l l y d i s t r i b u t e d w i t h m e a n

a n d v a r i a n c e

2

. I n s y m b o l s :

P

S

T

S

0

2 a ; b

= P

l o g

S

T

S

0

2 l o g a ; l o g b

=

Z

l o g b

l o g a

1

p

2

e x p

,

x ,

2

2

2

d x

N o t i c e t h a t s t o c k p r i c e s s h o u l d s t a y p o s i t i v e , s o a , b a r e p o s i t i v e .

O u r r s t g u e s s m i g h t b e t h a t E S

T

s h o u l d r e p r e s e n t a f a i r p r i c e t o w r i t e i n t o

o u r c o n t r a c t . H o w e v e r , i t w o u l d b e a r a r e c o i n c i d e n c e f o r t h i s t o b e t h e m a r k e t

p r i c e .

F o r s i m p l i c i t y , w e a l w a y s a s s u m e t h a t m a r k e t p a r t i c i p a n t s c a n b o r r o w m o n e y

f o r t h e s a m e r i s k - f r e e r a t e o f i n t e r e s t a s t h e y c a n l e n d m o n e y . T h u s , i f I b o r r o w $ 1

n o w , m y d e b t a t t i m e T w i l l b e $ e

r

, s a y , a n d s i m i l a r l y i f I k e e p $ 1 i n t h e b a n k t h e n

a t t i m e T i t w i l l b e w o r t h $ e

r

. S e e H u l l p . 5 0 f o r a d i s c u s s i o n o f t h i s a s s u m p t i o n .

W e n o w s h o w t h a t i t i s t h e i n t e r e s t r a t e a n d n o t o u r l o g - n o r m a l m o d e l t h a t f o r c e s

a c h o i c e o f K u p o n u s .

1 . S u p p o s e r s t t h a t K S

0

e

r

. T h e n m y c o u n t e r p a r t c a n a d o p t t h e f o l l o w i n g

s t r a t e g y : s h e b o r r o w s $ S

0

a t t i m e z e r o a n d b u y s w i t h i t o n e u n i t o f s t o c k .

A t t i m e T , s h e m u s t r e p a y $ S

0

e

r

t o t h e b a n k , b u t s h e h a s t h e s t o c k t o s e l l

t o m e f o r $ K , l e a v i n g h e r a c e r t a i n p r o t o f $ S

0

e

r

, K .



2 . I f K S

0

e

r

, t h e n I c a n r e v e r s e t h i s s t r a t e g y . I s e l l a u n i t o f s t o c k a t t i m e

z e r o f o r $ S

0

a n d I p u t t h e m o n e y i n t h e b a n k . A t t i m e T , I h a v e a c c u m u l a t e d

$ S

0

e

r

a n d I u s e $ K t o b u y a u n i t o f s t o c k l e a v i n g m e w i t h a c e r t a i n p r o t

o f $ S

0

e

r

, K .

U n l e s s K = S

0

e

r

, o n e o f u s i s g u a r a n t e e d t o m a k e a p r o t .

R e m a r k . I n t h i s a r g u m e n t , I s o l d s t o c k t h a t I m a y n o t o w n . T h i s i s k n o w n a s

s h o r t s e l l i n g . T h i s c a n , a n d d o e s , h a p p e n : i n v e s t o r s c a n h o l d n e g a t i v e a m o u n t s

o f s t o c k . F o r t h e d e t a i l s o f t h e m e c h a n i s m s i n v o l v e d s e e H u l l p . 4 8 - 9 .

D e n i t i o n 1 . 4 A n o p p o r t u n i t y t o l o c k i n t o a r i s k f r e e p r o t i s c a l l e d a n a r b i t r a g e

o p p o r t u n i t y .

T h e s t a r t i n g p o i n t i n e s t a b l i s h i n g a m o d e l i n m o d e r n n a n c e t h e o r y i s t o s p e c i f y

t h a t t h e r e a r e n o a r b i t r a g e o p p o r t u n i t i e s . I n f a c t t h e r e a r e p e o p l e w h o m a k e t h e i r

l i v i n g e n t i r e l y f r o m e x p l o i t i n g a r b i t r a g e o p p o r t u n i t i e s , b u t s u c h o p p o r t u n i t i e s d o

n o t e x i s t f o r a s i g n i c a n t l e n g t h o f t i m e b e f o r e m a r k e t p r i c e s m o v e t o e l i m i n a t e

t h e m .

O f c o u r s e f o r w a r d s a r e a v e r y s p e c i a l s o r t o f d e r i v a t i v e . T h e a r g u m e n t a b o v e

w o n ' t t e l l u s h o w t o v a l u e a n o p t i o n , b u t t h e s t r a t e g y o f s e e k i n g a p r i c e t h a t d o e s

n o t p r o v i d e e i t h e r p a r t y w i t h a n a r b i t r a g e o p p o r t u n i t y w i l l b e f u n d a m e n t a l i n

w h a t f o l l o w s .

2 D i s c r e t e t i m e m o d e l s I

2 . 1 T h e s i n g l e p e r i o d b i n a r y m o d e l

F o r m o s t o f w h a t f o l l o w s w e s h a l l b e i n t e r e s t e d i n n d i n g t h e ` f a i r ' p r i c e f o r a

E u r o p e a n c a l l o p t i o n . W e b e g i n w i t h a s i m p l e e x a m p l e .

E x a m p l e 2 . 1 S u p p o s e t h a t t h e c u r r e n t p r i c e i n S w i s s F r a n c s S F R o f $ 1 0 0 i s

S

0

= 1 5 0 . C o n s i d e r a E u r o p e a n c a l l o p t i o n w i t h s t r i k e p r i c e K = 1 5 0 a t t i m e T .

W h a t i s a f a i r p r i c e t o p a y f o r t h i s o p t i o n ?

R e c a l l t h a t s u c h a n o p t i o n g i v e s t h e b u y e r t h e r i g h t t o b u y $ 1 0 0 f o r 1 5 0 S F R a t

t i m e T . N o w t h e t r u e e x c h a n g e r a t e a t t i m e T i s n o t s o m e t h i n g w e k n o w , b u t i t

c a n b e m o d e l l e d b y a r a n d o m v a r i a b l e . T h e s i m p l e s t m o d e l i s t h e s i n g l e p e r i o d

b i n a r y m o d e l w h e r e S

T

w h i c h h e r e d e n o t e s t h e c o s t i n S F R o f $ 1 0 0 a t t i m e T i s

a s s u m e d t o t a k e o n e o f t w o v a l u e s w i t h s p e c i e d p r o b a b i l i t i e s . L e t ' s s u p p o s e

S

T

=

1 8 0 w i t h p r o b a b i l i t y p

9 0 w i t h p r o b a b i l i t y 1 , p

T h e p a y o o f t h e o p t i o n w i l l b e 3 0 = 1 8 0 - 1 5 0 S F R w i t h p r o b a b i l i t y p a n d 0 w i t h

p r o b a b i l i t y 1 , p a n d s o h a s e x p e c t a t i o n 3 0 p S F R . I s 3 0 p S F R a f a i r p r i c e t o p a y

f o r t h e o p t i o n ?

A s s u m e f o r s i m p l i c i t y t h a t i n t e r e s t r a t e s a r e z e r o a n d t h a t c u r r e n c y i s b o u g h t a n d

s o l d a t t h e s a m e e x c h a n g e r a t e .



T o m a k e t h i n g s m o r e c o n c r e t e , s u p p o s e t h a t p = 0 : 5 . T h a t i s , w e a r e m o d e l l i n g

S

T

a s a r a n d o m v a r i a b l e t h a t t a k e s t h e v a l u e s 9 0 S F R a n d 1 8 0 S F R w i t h e q u a l

p r o b a b i l i t y . W e a r e t h e n a s k i n g w h e t h e r 1 5 S F R i s a f a i r p r i c e t o p a y f o r t h e

o p t i o n .

A s i n x 1 , b y ` f a i r ' w e m e a n t h a t t h e r e i s n o p o s s i b i l i t y o f a r i s k f r e e p r o t f o r

e i t h e r t h e b u y e r o r t h e s e l l e r o f t h e o p t i o n . I c l a i m t h a t i f t h e p r i c e i s 1 5 S F R ,

t h e n I c a n m a k e a r i s k - f r e e p r o t b y t h e f o l l o w i n g s t r a t e g y : I b u y t h e o p t i o n a n d

I b o r r o w $ 3 3 . 3 3 a n d c o n v e r t i t s t r a i g h t i n t o 5 0 S F R . I s h a l l h a v e t o p a y o m y

l o a n i n d o l l a r s a t t i m e T .

M y p o s i t i o n a t t i m e z e r o i s a s f o l l o w s : I h a v e o n e o p t i o n t o b u y $ 1 0 0 f o r

1 5 0 S F R a t t i m e T , I h a v e 3 5 S F R 5 0 f r o m t h e c o n v e r s i o n o f m y d o l l a r l o a n , l e s s

1 5 p a i d f o r t h e o p t i o n a n d I h a v e a d e b t o f $ 3 3 . 3 3 .

A t t i m e T , o n e o f t w o t h i n g s h a s h a p p e n e d :

1 . I f S

T

= 1 8 0 , t h e n I e x e r c i s e t h e o p t i o n a n d b u y $ 1 0 0 f o r 1 5 0 S F R . I u s e

$ 3 3 . 3 3 t o p a y o m y d o l l a r d e b t , l e a v i n g m e $ 6 6 . 6 7 . T h i s I c o n v e r t b a c k i n t o

S F R a t t h e c u r r e n t e x c h a n g e r a t e . T h i s n e t s 2 = 3 1 8 0 = 1 2 0 S F R . I n t o t a l

t h e n , i f S

T

= 1 8 0 , a t t i m e T I h a v e 3 5 S F R - 1 5 0 S F R u s e d t o e x e r c i s e t h e

o p t i o n + 1 2 0 S F R , w h i c h t o t a l s 5 S F R c l e a r p r o t .

2 . I f S

T

= 9 0 , t h e n I t h r o w a w a y t h e o p t i o n w h i c h i s w o r t h l e s s a n d c o n v e r t

m y 3 5 S F R i n t o d o l l a r s , n e t t i n g $ 0 : 9 3 5 = $ 3 8 : 8 9 . I p a y o m y d e b t l e a v i n g

a p r o t o f $ 5 . 5 6 .

W h a t e v e r t h e t r u e e x c h a n g e r a t e a t t i m e T , I m a k e a p r o t .

S o t h e p r i c e o f t h e o p t i o n i s t o o l o w a t l e a s t f r o m t h e p o i n t o f v i e w o f t h e s e l l e r .

W h a t i s t h e r i g h t p r i c e ?

L e t ' s t h i n k o f t h i n g s f r o m t h e p o i n t o f v i e w o f t h e s e l l e r . I f I a m t h e s e l l e r o f

t h e o p t i o n , t h e n I k n o w t h a t a t t i m e T , I s h a l l n e e d $ S

T

, 1 5 0

+

i n o r d e r t o m e e t

t h e c l a i m a g a i n s t m e . T h e i d e a i s t o c a l c u l a t e h o w m u c h m o n e y I n e e d a t t i m e

z e r o t o b e h e l d i n a c o m b i n a t i o n o f d o l l a r s a n d S w i s s F r a n c s t o g u a r a n t e e t h i s .

S u p p o s e t h e n I h o l d $ x

1

a n d x

2

S F R a t t i m e z e r o . I n e e d t h i s h o l d i n g t o b e

w o r t h a t l e a s t S

T

, 1 5 0

+

S F R a t t i m e T .

1 . I f S

T

= 1 8 0 , t h e n I s h a l l n e e d a t l e a s t 3 0 S F R . T h a t i s , w e m u s t h a v e

x

1

+

1 8 0

1 0 0

x

2

3 0 :

2 . O n t h e o t h e r h a n d , i f S

T

= 9 0 , t h e n t h e p a y o o f t h e o p t i o n i s z e r o a n d I

j u s t n e e d n o t t o b e o u t o f p o c k e t . T h a t i s , I w a n t

x

1

+

9 0

1 0 0

x

2

0 :

A p r o t i s g u a r a n t e e d w i t h o u t r i s k f o r t h e s e l l e r i f x

1

; x

2

l i e s i n t h e i n t e r i o r

o f t h e s h a d e d r e g i o n i n F i g u r e 1 . O n t h e b o u n d a r y o f t h e r e g i o n , t h e r e i s a

p o s i t i v e p r o b a b i l i t y o f p r o t a n d n o p r o b a b i l i t y o f l o s s a t a l l p o i n t s o t h e r t h a n

t h e i n t e r s e c t i o n o f t h e t w o l i n e s . A t t h a t p o i n t t h e s e l l e r i s g u a r a n t e e d t o h a v e

e x a c t l y t h a t m o n e y r e q u i r e d t o m e e t t h e c l a i m a g a i n s t h e r a t t i m e T .



0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

x + 0.9x =0

2

21

x + 1.8 x =301

x

x2

1

F i g u r e 1 .

S o l v i n g t h e s i m u l t a n e o u s e q u a t i o n s g i v e s t h a t t h e s e l l e r c a n e x a c t l y m e e t t h e

c l a i m i f x

1

=

, 3 0 a n d x

2

= 3 0 0 = 9 . N o w t o p u r c h a s e $ 3 0 0 = 9 a t t i m e z e r o w o u l d

r e q u i r e 3 0 0 = 9 1 5 0 = 1 0 0 = 5 0 S F R . t h e v a l u e o f t h e p o r t f o l i o a t t i m e z e r o i s t h e n

5 0 , 3 0 = 2 0 S F R . T h e s e l l e r r e q u i r e s e x a c t l y 2 0 S F R a t t i m e z e r o t o c o n s t r u c t a

p o r t f o l i o t h a t w i l l b e w o r t h e x a c t l y t h e p a y o o f t h e o p t i o n a t t i m e T .

O n e c a n r e v e r s e t h e a r g u m e n t a b o v e t o s h o w t h a t f o r a n y l o w e r p r i c e , t h e r e i s

a s t r a t e g y f o r w h i c h t h e b u y e r m a k e s a r i s k - f r e e p r o t .

T h e f a i r p r i c e i s 2 0 S F R .

N o t i c e t h a t w e d i d n o t u s e t h e p r o b a b i l i t y , p , o f t h e p r i c e S

T

g o i n g u p t o

1 8 0 S F R a t a n y p o i n t i n t h e c a l c u l a t i o n . W e j u s t n e e d e d t h e f a c t t h a t w e c o u l d

r e p l i c a t e t h e c l a i m b y t h i s s i m p l e p o r t f o l i o . T h e s e l l e r c a n h e d g e t h e c o n t i n g e n t

c l a i m S

T

, 1 5 0

+

u s i n g t h e p o r t f o l i o c o n s i s t i n g o f x

1

S F R a n d $ x

2

.

O n e c a n u s e t h e s a m e a r g u m e n t t o p r o v e t h e f o l l o w i n g r e s u l t .

L e m m a 2 . 2 S u p p o s e t h a t t h e r i s k f r e e i n t e r e s t r a t e i s s u c h t h a t $ 1 d e p o s i t e d n o w

w i l l b e w o r t h $ e

r T

a t t i m e T . D e n o t e t h e t i m e z e r o v a l u e o f a c e r t a i n a s s e t b y

S

0

. S u p p o s e t h a t t h e m o t i o n o f s t o c k p r i c e s i s s u c h t h a t t h e v a l u e o f t h e a s s e t a t

t i m e T w i l l b e e i t h e r S

0

u o r S

0

d . A s s u m e f u r t h e r t h a t

d e

r T

u :

A t t i m e z e r o , t h e m a r k e t p r i c e o f a E u r o p e a n c a l l o p t i o n b a s e d o n t h i s s t o c k w i t h

s t r i k e p r i c e K a n d m a t u r i t y T i s

1 , d e

, r t

u , d

S

0

u , K

+

+

u e

, r t

, 1

u , d

S

0

d , K

+

:

T h e p r o o f i s e x e r c i s e 1 o n t h e p r o b l e m s h e e t .

R e m a r k 2 . 3 A t e r n a r y m o d e l .

T h e r e w e r e s e v e r a l t h i n g s a b o u t t h e b i n a r y m o d e l t h a t w e r e v e r y s p e c i a l . I n p a r -

t i c u l a r w e a s s u m e d t h a t w e k n e w t h a t t h e a s s e t p r i c e w o u l d b e o n e o f j u s t t w o

s p e c i e d v a l u e s a t t i m e T . W h a t i f w e a l l o w t h r e e v a l u e s ?

W e c a n t r y t o r e p e a t t h e a n a l y s i s a b o v e . A g a i n t h e s e l l e r w o u l d l i k e t o r e p l i c a t e

t h e c l a i m a t t i m e T b y a p o r t f o l i o c o n s i s t i n g o f x

1

S F R a n d $ x

2

. T h i s t i m e t h e r e



w i l l b e t h r e e s c e n a r i o s t o c o n s i d e r , c o r r e s p o n d i n g t o t h e t h r e e p o s s i b l e v a l u e s o f

S

T

. T h i s g i v e s r i s e t o t h r e e i n e q u a l i t i e s :

x

1

+

S

i

T

1 0 0

x

2

S

i

T

, 1 5 0

+

i = 1 ; 2 ; 3 ;

w h e r e S

i

T

a r e t h e p o s s i b l e v a l u e s o f S

T

s e e F i g u r e 2 .

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 01 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 01 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1x

x

2

1

F i g u r e 2 .

I n o r d e r t o b e g u a r a n t e e d t o m e e t t h e c l a i m a t t i m e T , t h e s e l l e r r e q u i r e s x

1

; x

2

t o l i e i n t h e s h a d e d r e g i o n , b u t a t a n y p o i n t i n t h a t r e g i o n , s h e h a s a p o s i t i v e

p r o b a b i l i t y o f m a k i n g a p r o t a n d z e r o p r o b a b i l i t y o f m a k i n g a l o s s . T h e r e i s n o

p o r t f o l i o t h a t e x a c t l y r e p l i c a t e s t h e c l a i m a n d t h e r e i s n o u n i q u e ` f a i r ' p r i c e f o r

t h e o p t i o n .

O u r m a r k e t i s n o t c o m p l e t e . T h a t i s , t h e r e a r e c o n t i n g e n t c l a i m s t h a t c a n n o t

b e p e r f e c t l y h e d g e d .

I f w e a l l o w e d t h e s e l l e r t o t r a d e i n a t h i r d ` i n d e p e n d e n t ' a s s e t , t h e n o u r a r g u -

m e n t w o u l d l e a d u s t o t h r e e l i n e a r e q u a t i o n s i n t h r e e u n k n o w n s , c o r r e s p o n d i n g

t o t h r e e n o n - p a r a l l e l p l a n e s i n R

3

w h i c h i n t e r s e c t i n a s i n g l e p o i n t . T h a t p o i n t

r e p r e s e n t s t h e u n i q u e p o r t f o l i o t h a t e x a c t l y r e p l i c a t e s t h e c l a i m a t t i m e T .

2 . 2 A c h a r a c t e r i s a t i o n o f ` n o a r b i t r a g e '

I n o u r b i n a r y s e t t i n g i t w a s e a s y t o n d t h e r i g h t p r i c e f o r o u r o p t i o n b y s o l v i n g

a p a i r o f s i m u l t a n e o u s e q u a t i o n s i n t w o u n k n o w n s . I n m o r e c o m p l e x s i t u a t i o n s ,

i t i s c o n v e n i e n t t o h a v e a r a t h e r d i e r e n t v i e w p o i n t . T h i s n e w v i e w p o i n t w i l l a l s o

c a r r y o v e r t o o u r c o n t i n u o u s m o d e l s a n d i s p o s s i b l y t h e s i n g l e m o s t i m p o r t a n t

i d e a i n v a l u a t i o n o f d e r i v a t i v e s . T h i s w i l l t a k e u s b a c k i n t o p r o b a b i l i t y t h e o r y ,

b u t r s t w e n e e d a c o n c i s e m a t h e m a t i c a l c o n d i t i o n t o c h a r a c t e r i s e m a r k e t s t h a t

h a v e n o a r b i t r a g e o p p o r t u n i t i e s .

O u r m a r k e t w i l l n o w c o n s i s t o f a n i t e b u t p o s s i b l y l a r g e n u m b e r o f t r a d e a b l e

a s s e t s .

S u p p o s e t h e n t h a t t h e r e a r e N t r a d e a b l e a s s e t s i n t h e m a r k e t . T h e i r p r i c e s

a t t i m e z e r o a r e g i v e n b y t h e c o l u m n v e c t o r S

0

= S

1

0

; S

2

0

; : : : ; S

N

0

T

. U n c e r t a i n t y

a b o u t t h e m a r k e t i s r e p r e s e n t e d b y a n i t e n u m b e r o f p o s s i b l e s t a t e s i n w h i c h t h e

m a r k e t m i g h t b e a t t i m e o n e t h a t w e l a b e l 1 ; 2 ; : : : ; n . T h e s e c u r i t y v a l u e s a t t i m e



o n e a r e g i v e n b y a n N n m a t r i x D = D

i j

, w h e r e t h e c o e c i e n t D

i j

i s t h e v a l u e

o f t h e i t h s e c u r i t y a t t i m e o n e i f t h e m a r k e t i s i n s t a t e j .

I n t h i s n o t a t i o n , a p o r t f o l i o c a n b e t h o u g h t o f a s a v e c t o r =

1

;

2

; : : : ;

n

T

2

R

N

, w h o s e m a r k e t v a l u e a t t i m e z e r o i s t h e s c a l a r p r o d u c t S

0

: = S

1

0

1

+ S

2

0

2

+

: : : + S

N

0

N

. T h e p a y o o f t h e p o r t f o l i o a t t i m e o n e i s a v e c t o r i n R

n

w h o s e i t h

e n t r y i s t h e v a l u e o f t h e p o r t f o l i o i f t h e m a r k e t i s i n s t a t e i . W e c a n w r i t e t h e

p a y o a s

0

B

B

@

D

1 1

1

+ D

2 1

2

+ D

N 1

N

D

1 2

1

+ D

2 2

2

+ D

N 2

N

.

.

.

D

1 n

1

+ D

2 n

2

+ D

N n

N

1

C

C

A

= D

T

:

N o t a t i o n

F o r a v e c t o r x 2 R

n

w e w r i t e x 0 t o m e a n x 2 R

n

+

a n d x 0 t o m e a n

x 0 ; x 6 = 0 . N o t i c e t h a t x 0 d o e s n o t r e q u i r e x t o b e s t r i c t l y p o s i t i v e i n

a l l i t s c o o r d i n a t e s . W e w r i t e x 0 , f o r v e c t o r s w h i c h a r e s t r i c t l y p o s i t i v e i n a l l

c o o r d i n a t e s , i e f o r v e c t o r s x 2 R

n

+ +

.

A n a r b i t r a g e i s t h e n a p o r t f o l i o 2 R

N

w i t h

S

0

: 0 ; D

T

0 o r S

0

: 0 ; D

T

0 :

D e n i t i o n 2 . 4 A s t a t e p r i c e v e c t o r i s a v e c t o r 2 R

n

+ +

s u c h t h a t S

0

= D .

E x p a n d i n g t h i s g i v e s

0

B

B

@

S

1

0

S

2

0

.

.

.

S

N

0

1

C

C

A

=

1

0

B

B

@

D

1 1

D

2 1

.

.

.

D

N 1

1

C

C

A

+

2

0

B

B

@

D

1 2

D

2 2

.

.

.

D

N 2

1

C

C

A

+ +

n

0

B

B

@

D

1 n

D

2 n

.

.

.

D

N n

1

C

C

A

:

T h e v e c t o r m u l t i p l i e d b y

i

i s t h e s e c u r i t y p r i c e v e c t o r i f t h e m a r k e t i s i n s t a t e i .

W e c a n t h i n k o f

i

a s t h e m a r g i n a l c o s t o f o b t a i n i n g a n a d d i t i o n a l u n i t o f w e a l t h

a t t h e e n d o f t h e t i m e p e r i o d i f t h e s y s t e m i s i n s t a t e i .

T h e o r e m 2 . 5 T h e r e i s n o a r b i t r a g e i f a n d o n l y i f t h e r e i s a s t a t e p r i c e v e c t o r .

T h e p r o o f i s a n a p p l i c a t i o n o f t h e f o l l o w i n g r e s u l t f r o m c o n v e x i t y t h e o r y f o r a

d i s c u s s i o n s e e A p p e n d i x B o f D u e a n d t h e r e f e r e n c e s t h e r e i n . R e c a l l t h a t

M R

d

i s a c o n e i f x 2 M i m p l i e s x 2 M f o r a l l s t r i c t l y p o s i t i v e s c a l a r s .

T h e o r e m 2 . 6 S u p p o s e M a n d K a r e c l o s e d c o n v e x c o n e s i n R

d

t h a t i n t e r s e c t

p r e c i s e l y a t t h e o r i g i n . I f K i s n o t a l i n e a r s u b s p a c e , t h e n t h e r e i s a n o n - z e r o

l i n e a r f u n c t i o n a l F s u c h t h a t F x F y f o r e a c h x 2 M a n d e a c h n o n - z e r o

y 2 K .

R e m i n d e r : A l i n e a r f u n c t i o n a l o n R

d

i s a l i n e a r m a p p i n g F : R

d

! R . B y

t h e R i e s z R e p r e s e n t a t i o n T h e o r e m , a n y b o u n d e d l i n e a r f u n c t i o n a l o n R

d

c a n b e



w r i t t e n a s F x = v

0

: x . T h a t i s F x i s t h e s c a l a r ` d o t ' p r o d u c t o f s o m e x e d

v e c t o r v

0

2 R

d

w i t h x .

P r o o f o f T h e o r e m 2 . 5

W e t a k e d = 1 + n i n T h e o r e m 2 . 6 a n d s e t

M =

,

, S

0

: ; D

T

: 2 R

N

R R

n

= R

1 + n

;

K = R

+

R

n

+

:

N o t e t h a t K i s a c o n e a n d M i s a l i n e a r s p a c e .

0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0

1 1 1 1 1 1 1 1 1 1 1 1

1 1 1 1 1 1 1 1 1 1 1 1

1 1 1 1 1 1 1 1 1 1 1 1

1 1 1 1 1 1 1 1 1 1 1 1

1 1 1 1 1 1 1 1 1 1 1 1

1 1 1 1 1 1 1 1 1 1 1 1

1 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 1

R

R

1

n

M

K

F i g u r e 3 .

E v i d e n t l y , t h e r e i s n o a r b i t r a g e i f a n d o n l y i f K a n d M i n t e r s e c t p r e c i s e l y a t t h e

o r i g i n a s s h o w n i n F i g u r e 3 . W e m u s t p r o v e t h a t K M = f 0 g i f a n d o n l y i f t h e r e

i s a s t a t e p r i c e v e c t o r .

i S u p p o s e r s t t h a t K M = f 0 g . F r o m T h e o r e m 2 . 6 , t h e r e i s a l i n e a r

f u n c t i o n a l F : R

d

! R s u c h t h a t F z F x f o r a l l z 2 M a n d n o n - z e r o x 2 K .

T h e r s t s t e p i s t o s h o w t h a t F m u s t v a n i s h o n M . W e e x p l o i t t h e f a c t t h a t M

i s a l i n e a r s p a c e .

F i r s t o b s e r v e t h a t F 0 = 0 b y l i n e a r i t y o f F a n d s o F x 0 f o r x 2 K a n d

F x 0 f o r x 2 K n f 0 g . F i x x

0

2 K w i t h x

0

6 = 0 . N o w t a k e a n a r b i t r a r y z 2 M .

T h e n F z F x

0

, b u t a l s o , s i n c e M i s a l i n e a r s p a c e , F z = F z F x

0

f o r a l l 2 R . T h i s c a n o n l y h o l d i f F z = 0 . z 2 M w a s a r b i t r a r y a n d s o F

v a n i s h e s o n M a s r e q u i r e d .

W e n o w u s e t h i s t o a c t u a l l y e x p l i c i t l y c o n s t r u c t t h e s t a t e p r i c e v e c t o r f r o m

F . F i r s t w e u s e t h e R i e s z R e p r e s e n t a t i o n T h e o r e m t o w r i t e F a s F x = v

0

: x f o r

s o m e v

0

2 R

d

. I t i s c o n v e n i e n t t o w r i t e v

0

= ; f o r s o m e 2 R , 2 R

n

. T h e n

F v ; c = v + : c f o r a n y v ; c 2 R R

n

= R

d

:



S i n c e F x 0 f o r a l l n o n - z e r o x 2 K , w e m u s t h a v e 0 a n d 0 c o n s i d e r

a v e c t o r a l o n g e a c h o f t h e c o - o r d i n a t e a x e s . F i n a l l y , s i n c e F v a n i s h e s o n M ,

, S

0

: + : D

T

= 0 8 2 R

N

:

O b s e r v i n g t h a t : D

T

= D : , t h i s b e c o m e s

, S

0

: + D : = 0 8 2 R

N

;

w h i c h i m p l i e s t h a t S

0

= D = . T h e v e c t o r = = i s a s t a t e p r i c e v e c t o r .

i i S u p p o s e n o w t h a t t h e r e i s a s t a t e p r i c e v e c t o r . W e m u s t p r o v e t h a t K M =

f 0 g .

W r i t i n g f o r t h e s t a t e p r i c e v e c t o r , S

0

= D . T h e n f o r a n y p o r t f o l i o ,

S

0

: = D : = : D

T

:

N o w , i f D

T

2 R

n

+

, t h e n , s i n c e 0 , : D

T

0 . I f , S

0

: ; D

T

2 K ,

D

T

2 R

n

+

a n d

, S

0

:

0 . T h a t i s , w e m u s t h a v e : D

T

0 a n d

, S

0

: =

, : D

T

0 . T h i s h a p p e n s i f a n d o n l y i f , S

0

: = : D

T

= 0 . T h a t i s ,

K

M =

f 0

g , a s r e q u i r e d . 2

2 . 3 T h e r i s k n e u t r a l p r o b a b i l i t y m e a s u r e .

I n o r d e r t o g e t b a c k t o p r o b a b i l i t y t h e o r y , w e a r e g o i n g t o t h i n k o f t h e s t a t e

p r i c e v e c t o r r a t h e r d i e r e n t l y . R e c a l l t h a t a l l t h e e n t r i e s o f a r e s t r i c t l y p o s i t i v e .

W r i t i n g

0

=

P

n

i = 1

i

, w e c a n t h i n k o f

=

1

0

;

2

0

; : : : ;

n

0

T

1

a s a v e c t o r o f p r o b a b i l i t i e s f o r b e i n g i n d i e r e n t s t a t e s . O f c o u r s e t h e y m a y h a v e

n o t h i n g t o d o w i t h o u r v i e w o f h o w t h e m a r k e t s w i l l m o v e .

F i r s t o f a l l , w h a t i s

0

?

S u p p o s e t h a t t h e m a r k e t a l l o w s p o s i t i v e r i s k l e s s b o r r o w i n g , b y w h i c h w e m e a n

t h a t f o r s o m e p o r t f o l i o , ,

D

T

=

0

B

B

@

1

1

.

.

.

1

1

C

C

A

;

i . e . t h e v a l u e o f t h e p o r t f o l i o a t t i m e T i s o n e , n o m a t t e r w h a t s t a t e t h e m a r k e t

i s i n . U s i n g t h e f a c t t h a t i s a s t a t e p r i c e v e c t o r , w e c a l c u l a t e t h a t t h e c o s t o f

s u c h a p o r t f o l i o a t t i m e z e r o i s

S

0

: = D : = : D

T

=

n

X

i = 1

i

=

0

:

T h a t i s

0

r e p r e s e n t s t h e d i s c o u n t o n r i s k l e s s b o r r o w i n g .



N o w u n d e r t h e p r o b a b i l i t y d i s t r i b u t i o n g i v e n b y t h e v e c t o r 1 , t h e e x p e c t e d

v a l u e o f t h e i t h s e c u r i t y a t t i m e T i s

E

S

i

T

=

n

X

j = 1

D

i j

j

0

=

1

0

n

X

j = 1

D

i j

j

=

1

0

S

i

0

;

w h e r e i n t h e l a s t e q u a l i t y w e h a v e u s e d S

0

= D . T h a t i s

S

i

0

=

0

E

S

i

T

; i = 1 ; : : : ; n : 2

A n y s e c u r i t y ' s p r i c e i s i t s d i s c o u n t e d e x p e c t e d p a y o u n d e r t h e p r o b a b i l i t y d i s t r i -

b u t i o n 1 .

T h i s o b s e r v a t i o n g i v e s u s a n e w w a y t o t h i n k a b o u t t h e p r i c i n g o f c o n t i n g e n t

c l a i m s . W e s h a l l s a y t h a t a c l a i m i s a t t a i n a b l e i f i t c a n b e h e d g e d . T h a t i s , i f

t h e r e i s a p o r t f o l i o w h o s e v a l u e a t t i m e T i s e x a c t l y C .

T h e o r e m 2 . 7 I f t h e r e i s n o a r b i t r a g e , t h e u n i q u e t i m e z e r o p r i c e o f a n a t t a i n a b l e

c l a i m C a t t i m e T i s

0

E C w h e r e t h e e x p e c t a t i o n i s w i t h r e s p e c t t o a n y p r o b a -

b i l i t y m e a s u r e f o r w h i c h S

i

0

=

0

E S

i

T

f o r a l l i a n d

0

i s t h e d i s c o u n t o n r i s k l e s s

b o r r o w i n g .

R e m a r k . T h e s a m e v a l u e i s o b t a i n e d i f t h e e x p e c t a t i o n i s c a l c u l a t e d f o r a n y

v e c t o r o f p r o b a b i l i t i e s s u c h t h a t S

i

0

=

0

E S

i

T

s i n c e , i n t h e a b s e n c e o f a r b i t r a g e ,

t h e r e i s o n l y o n e r i s k l e s s b o r r o w i n g r a t e .

T h i s r e s u l t s a y s t h a t i f I c a n n d a p r o b a b i l i t y v e c t o r f o r w h i c h t h e v a l u e o f

e a c h s e c u r i t y n o w i s i t s d i s c o u n t e d e x p e c t e d v a l u e a t t i m e T t h e n I c a n n d t h e

t i m e z e r o v a l u e o f a n y a t t a i n a b l e c o n t i n g e n t c l a i m b y c a l c u l a t i n g t h e e x p e c t a t i o n .

I w i l l b e u s i n g t h e s a m e p r o b a b i l i t y v e c t o r , w h a t e v e r t h e c l a i m .

P r o o f o f T h e o r e m 2 . 7

S i n c e t h e c l a i m c a n b e h e d g e d , t h e r e i s a p o r t f o l i o s o t h a t : S

T

= C . N o w

: S

0

= :

0

E S

T

=

0

N

X

i = 1

i

E S

i

T

=

0

E : S

T

;

a s r e q u i r e d . 2

L e t ' s g o b a c k t o o u r v e r y r s t o p t i o n p r i c i n g e x a m p l e .

E x a m p l e 2 . 1 r e v i s i t e d .

R e c a l l t h e c o n d i t i o n s i n t h a t e x a m p l e : t h e c u r r e n t p r i c e i n S w i s s F r a n c s S F R

o f $ 1 0 0 i s S

0

= 1 5 0 . W e s u p p o s e t h a t a t t i m e T , t h e p r i c e w i l l h a v e m o v e d t o

e i t h e r 9 0 S F R o r 1 8 0 S F R . O u r p r o b l e m w a s t o p r i c e a E u r o p e a n c a l l o p t i o n w i t h

s t r i k e p r i c e K = 1 5 0 a t t i m e T . T h e h o l d e r o f s u c h a n o p t i o n h a s t h e r i g h t , b u t

n o t t h e o b l i g a t i o n , t o b u y $ 1 0 0 f o r 1 5 0 S F R a t t i m e T . F o r s i m p l i c i t y , i n t e r e s t

r a t e s w e r e a s s u m e d t o b e z e r o .

W e a l r e a d y k n o w t h e f a i r p r i c e f o r s u c h a n o p t i o n i s 2 0 S F R , b u t l e t s s e e h o w

t o o b t a i n t h i s w i t h o u r n e w a p p r o a c h .



U n d e r o u r a s s u m p t i o n o f z e r o i n t e r e s t r a t e s , w e h a v e

0

= 1 a n d s o w e a r e

s e e k i n g a p r o b a b i l i t y v e c t o r s u c h t h a t E S

T

= S

0

. T h a t i s , w e a r e l o o k i n g f o r a

p r o b a b i l i t y v e c t o r f o r w h i c h t h e e x c h a n g e r a t e b e h a v e s l i k e a f a i r g a m e . L e t p b e

t h e p r o b a b i l i t y t h a t S

T

= 1 8 0 S F R , t h e n s i n c e S

T

c a n o n l y t a k e v a l u e s 1 8 0 a n d 9 0

i t m u s t b e t h a t P S

T

= 9 0 = 1 , p . F o r t h e p r i c e t o b e h a v e l i k e a f a i r g a m e , w e

r e q u i r e t h a t

1 8 0 p + 9 0 1 , p = 1 5 0

w h i c h y i e l d s p = 2 = 3 . T h e c l a i m a t t i m e T i s C = S

T

, 1 5 0

+

, a n d s o u s i n g

T h e o r e m 2 . 7 , w e s e e t h a t t h e f a i r p r i c e i n S F R i s

E S

T

, 1 5 0

+

=

2

3

1 8 0 , 1 5 0

+

+

1

3

9 0 , 1 5 0

+

= 2 0 ;

a s b e f o r e .

A r m e d w i t h t h e p r o b a b i l i t y p , i t i s a t r i v i a l m a t t e r t o v a l u e o t h e r o p t i o n s , f o r

e x a m p l e a E u r o p e a n c a l l w i t h s t r i k e p r i c e 1 2 0 S F R i n s t e a d o f 1 5 0 S F R w o u l d b e

v a l u e d a t

E S

T

, 1 2 0

+

=

2

3

1 8 0 , 1 2 0

+

+

1

3

9 0 , 1 2 0

+

= 4 0 :

T h e p r o b a b i l i t y m e a s u r e t h a t a s s i g n s p r o b a b i l i t y 2 3 t o S

T

= 1 8 0 S F R a n d 1 3

t o S

T

= 9 0 S F R i s c a l l e d a n e q u i v a l e n t m a r t i n g a l e p r o b a b i l i t y f o r t h e d i s c o u n t e d

p r i c e p r o c e s s f S

0

,

0

S

T

g . T h e p r o b a b i l i t i e s a r e a l s o s o m e t i m e s c a l l e d t h e r i s k

n e u t r a l p r o b a b i l i t i e s .

Documents

1.Financial Derivatives and Discrete Time Models I